Abstract

A bending loss formula for optical fibers with an axially symmetric arbitrary-index profile is derived by approximating the refractive-index profile with a staircase function. The permissible bending radius R* defined for a given value of bending loss is derived. It is deduced that R* is nearly proportional to wavelength λ when the normalized frequency υ and the refractive-index difference Δ are fixed. The ratio of R* at two different values of υ depends only on the ratio of υ. The influence of an index dip and profile smoothing on R* is numerically evaluated.

© 1978 Optical Society of America

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References

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  1. J. Sakai, T. Kimura, Opt. Lett. 1, 169 (1977).
    [Crossref] [PubMed]
  2. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
  3. L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
    [Crossref]
  4. J. A. Arnaud, Bell Syst. Tech. J. 53, 1379 (1974).
  5. A. W. Snyder, I. White, D. J. Mitchell, Electron. Lett. 11, 332 (1975).
    [Crossref]
  6. D. Marcuse, J. Opt. Soc. Am. 66, 216 (1976).
    [Crossref]
  7. D. Marcuse, J. Opt. Soc. Am. 66, 311 (1976).
    [Crossref]
  8. E. F. Kuester, D. C. Chang, IEEE. J. Quantum Electron. QE-11, 903 (1975).
    [Crossref]
  9. D. C. Chang, E. F. Kuester, Radio Sci. 11, 449 (1976).
    [Crossref]
  10. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [Crossref] [PubMed]
  11. P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
    [Crossref]
  12. Y. Suematsu, K. Furuya, IEEE Trans. Microwave Theory Tech. MTT-20, 524 (1972).
    [Crossref]
  13. T. Tanaka, Y. Suematsu, Paper at Technical Group on Optics & Quantum Electonics, IECE of Japan, OQE 75-26, 39 (1975).
  14. W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
    [Crossref]

1977 (2)

J. Sakai, T. Kimura, Opt. Lett. 1, 169 (1977).
[Crossref] [PubMed]

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[Crossref]

1976 (3)

1975 (2)

E. F. Kuester, D. C. Chang, IEEE. J. Quantum Electron. QE-11, 903 (1975).
[Crossref]

A. W. Snyder, I. White, D. J. Mitchell, Electron. Lett. 11, 332 (1975).
[Crossref]

1974 (2)

L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
[Crossref]

J. A. Arnaud, Bell Syst. Tech. J. 53, 1379 (1974).

1972 (1)

Y. Suematsu, K. Furuya, IEEE Trans. Microwave Theory Tech. MTT-20, 524 (1972).
[Crossref]

1971 (1)

1970 (1)

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[Crossref]

1969 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Arnaud, J. A.

J. A. Arnaud, Bell Syst. Tech. J. 53, 1379 (1974).

Chan, K. B.

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[Crossref]

Chang, D. C.

D. C. Chang, E. F. Kuester, Radio Sci. 11, 449 (1976).
[Crossref]

E. F. Kuester, D. C. Chang, IEEE. J. Quantum Electron. QE-11, 903 (1975).
[Crossref]

Clarricoats, P. J. B.

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[Crossref]

Furuya, K.

Y. Suematsu, K. Furuya, IEEE Trans. Microwave Theory Tech. MTT-20, 524 (1972).
[Crossref]

Gambling, W. A.

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[Crossref]

Gloge, D.

Kimura, T.

Kuester, E. F.

D. C. Chang, E. F. Kuester, Radio Sci. 11, 449 (1976).
[Crossref]

E. F. Kuester, D. C. Chang, IEEE. J. Quantum Electron. QE-11, 903 (1975).
[Crossref]

Lewin, L.

L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Marcuse, D.

Matsumura, H.

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[Crossref]

Mitchell, D. J.

A. W. Snyder, I. White, D. J. Mitchell, Electron. Lett. 11, 332 (1975).
[Crossref]

Payne, D. N.

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[Crossref]

Sakai, J.

Snyder, A. W.

A. W. Snyder, I. White, D. J. Mitchell, Electron. Lett. 11, 332 (1975).
[Crossref]

Suematsu, Y.

Y. Suematsu, K. Furuya, IEEE Trans. Microwave Theory Tech. MTT-20, 524 (1972).
[Crossref]

T. Tanaka, Y. Suematsu, Paper at Technical Group on Optics & Quantum Electonics, IECE of Japan, OQE 75-26, 39 (1975).

Tanaka, T.

T. Tanaka, Y. Suematsu, Paper at Technical Group on Optics & Quantum Electonics, IECE of Japan, OQE 75-26, 39 (1975).

White, I.

A. W. Snyder, I. White, D. J. Mitchell, Electron. Lett. 11, 332 (1975).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

J. A. Arnaud, Bell Syst. Tech. J. 53, 1379 (1974).

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Electron. Lett. (3)

A. W. Snyder, I. White, D. J. Mitchell, Electron. Lett. 11, 332 (1975).
[Crossref]

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[Crossref]

W. A. Gambling, D. N. Payne, H. Matsumura, Electron. Lett. 13, 139 (1977).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

Y. Suematsu, K. Furuya, IEEE Trans. Microwave Theory Tech. MTT-20, 524 (1972).
[Crossref]

L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
[Crossref]

IEEE. J. Quantum Electron. (1)

E. F. Kuester, D. C. Chang, IEEE. J. Quantum Electron. QE-11, 903 (1975).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Radio Sci. (1)

D. C. Chang, E. F. Kuester, Radio Sci. 11, 449 (1976).
[Crossref]

Other (1)

T. Tanaka, Y. Suematsu, Paper at Technical Group on Optics & Quantum Electonics, IECE of Japan, OQE 75-26, 39 (1975).

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Figures (11)

Fig. 1
Fig. 1

Approximation of refractive-index profile with a staircase function. n0 is the index at the core center, and ne is the index of the cladding. The arrows of Qi and Gi indicate regions where propagation and boundary matrices are defined.

Fig. 2
Fig. 2

Classification of refractive-index profile for determining characteristic equations: (a) profile where index n0 at the core center agrees with the maximum index nm; (b) profile where n0 is smaller than nm.

Fig. 3
Fig. 3

Section of bent optical fiber with cylindrical coordinate system ( r ̂ , ϕ ̂ , ) , local Cartesian system (x,y,z), and local cylindrical system (r,θ,z). The z axis is along the toroidal axis, and the axis is perpendicular to the toroidal axis. R is the radius of curvature, and a is the core radius.

Fig. 4
Fig. 4

Lower limit υL of normalized frequency υ for bending loss formula of the LP01 mode. G = ΔR/a. α is a profile parameter of an α-power law-index distribution.

Fig. 5
Fig. 5

Refractive-index profiles for evaluating the influence of an index dip and profile smoothing on the bending loss: (a) M type profile with an index dip. = (nmn0)/(nmne); (b) index profile with profile smoothing. n(r) is given in Eq. (34). Profiles with υ = 2.405, λ = 1.25 μm, and Δ = 0.3% are also depicted for three different D values.

Fig. 6
Fig. 6

Index dip effect on bending loss for the LP01 mode. R* is the permissible bending radius at which loss exceeds 0.1 dB/km. υ = 2.405, λ = 1.25 μm, and Δ = 0.3%. δ and are parameters characterizing an index dip shown in Fig. 5 (a).

Fig. 7
Fig. 7

Profile smoothing effect on bending loss for the LP01 mode. υ = 2.405 and λ = 1.25 μm. D is a parameter characterizing profile smoothing defined in Eq. (34).

Fig. 8
Fig. 8

Relation between λ and R* for the LP01 mode. Power law profile parameter α = 2 and υ = υc1 = 3.518. υc1 is the LP11 mode cutoff.

Fig. 9
Fig. 9

Relation between Δ and R* for the LP01 mode. Profile parameter α = 2 and υ = υc1 = 3.518. υc1 is the LP11 mode cutoff.

Fig. 10
Fig. 10

Permissible bending radius ratio vs normalized frequency ratio for the LP01 mode in an α-power law-index profile. α = 2,∞. υc1 is the LP11 mode cutoff.

Fig. 11
Fig. 11

Permissible bending radius ratio vs normalized frequency ratio for the LP11 mode in an α-power law-index profile. α = 2, ∞. υc1 and υc2 are the LP11 and LP21 mode cutoffs, respectively.

Equations (56)

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2 α = P r / P g ,
[ j E z E y ] r = a i = U [ d 11 , d 12 d 21 , d 22 ] [ A i B i ] ,
U = exp [ j ( ω t β g z ) ] .
d 11 = ( κ i / 2 β g ) [ J ν + 1 sin ( ν + 1 ) θ + J ν 1 sin ( ν 1 ) θ ] , d 12 = ( κ i / 2 β g ) [ N ν + 1 sin ( ν + 1 ) θ + N ν 1 sin ( ν 1 ) θ ] , d 21 = J ν cos ν θ , d 22 = N ν cos ν θ , }
κ i = ( n i 2 k 2 β g 2 ) 1 / 2 ,
k = 2 π / λ ,
d 11 = ( γ i / 2 β g ) [ K ν + 1 sin ( ν + 1 ) θ K ν 1 sin ( ν 1 ) θ ] , d 12 = ( γ i / 2 β g ) [ I ν + 1 sin ( ν + 1 ) θ I ν 1 sin ( ν 1 ) θ ] , d 21 = K ν cos ν θ , d 22 = I ν cos ν θ , }
γ i = ( β g 2 n i 2 k 2 ) 1 / 2 .
[ z y ] r = a i = U [ κ i J ν + 1 / β g , κ i N ν + 1 / β g J ν , N ν ] [ A i B i ] ,
[ z y ] r = a i = U [ γ i K ν + 1 / β g , γ i I ν + 1 / β g K ν , I ν ] [ A i B i ] .
[ z y ] r = r 2 = Q i [ z y ] r = r 1 = [ q 11 , q 12 q 21 , q 22 ] [ z y ] r = r 1 .
a i r 1 , r 2 a i + 1 .
[ z y ] r = a i i th layer = G i [ z y ] r = a i ( i 1 ) th layer ,
G i = [ 1 , 0 0 , ( n i 1 / n i ) 2 ] .
[ z y ] r = a i = F i [ z y ] r = a 1 = [ f 11 , f 12 f 21 , f 22 ] [ z y ] r = a 1 ,
F i = G i ( Q i 1 G i 1 ) . . . ( Q 1 G 1 ) .
K ν ( γ e a N ) [ f 11 κ 0 J ν + 1 ( κ 0 a 1 ) / β g f 12 J ν ( κ 0 a 1 ) ] + γ e K ν + 1 ( γ e a N ) / β g [ f 21 κ 0 J ν + 1 ( κ 0 a 1 ) / β g f 22 J ν ( κ 0 a 1 ) ] = 0 ,
κ 0 = ( n 0 2 k 2 β g 2 ) 1 / 2 , γ e = ( β g 2 n e 2 k 2 ) 1 / 2 .
K ν ( γ e a N ) [ f 11 γ 0 I ν + 1 ( γ 0 a 1 ) / β g + f 12 I ν ( γ 0 a 1 ) ] + γ e K ν + 1 ( γ e a N ) / β g [ f 21 γ 0 I ν + 1 ( γ 0 a 1 ) / β g + f 22 I ν ( γ 0 a 1 ) ] = 0 ,
γ 0 = ( β g 2 n 0 2 k 2 ) 1 / 2 , γ e = ( β g 2 n e 2 k 2 ) 1 / 2 .
P g = 1 2 0 2 π 0 ( E x H y * E y H x * ) r d r d θ = π s β g 2 k ( 0 μ 0 ) 1 / 2 P ,
s = { 2 ; ν = 0 , 1 ; otherwise ,
P = i P i .
P i = [ A i 2 r 2 2 ( J ν 2 J ν 1 J ν + 1 ) + B i 2 r 2 2 ( N ν 2 N ν 1 N ν + 1 ) + A i B i r 2 ( J ν N ν J ν 1 N ν + 1 ) ] a i a i + 1 .
P i = [ A i 2 r 2 2 ( K ν 2 K ν 1 K ν + 1 ) + B i 2 r 2 2 ( I ν 2 I ν 1 I ν + 1 ) + A i B i r 2 ( K ν I ν + K ν 1 I ν + 1 ) ] a i a i + 1 .
E = A e K ν ( γ e r ) cos ( ν θ ) exp ( j β g R ϕ ̂ ) ,
E ϕ ̂ = j A e γ e 2 β g [ K ν + 1 ( γ e r ) sin ( ν + 1 ) θ K ν 1 ( γ e r ) sin ( ν 1 ) θ ] exp ( j β g R ϕ ̂ ) ,
γ e = ( β g 2 n e 2 k 2 ) 1 / 2 .
c 1 ( β ) = j A e B 1 * 2 π N μ β 1 ω n e 2 0 ρ γ e ( β g β γ e + γ e ) H μ ( 2 ) * [ ρ ( R + a ) ] G ν β ,
c 2 ( β ) = j A e B 2 * 2 π N μ β 2 ω n e 2 0 ρ H μ ( 2 ) * [ ρ ( R + a ) ] G ν β ,
G ν β = K ν [ γ e ( a 2 + z 2 ) 1 / 2 ] × cos ν [ tan 1 ( z / a ) ] exp ( j β z ) d z .
P r = π r R i = 1 2 | c i ( β ) | 2 ( ϕ μ i μ i * μ i ϕ μ i * ) d β .
2 α = A e 2 P g ω n e 2 0 γ e 4 π ( n e k ) 2 exp ( 2 γ e a ) exp ( 2 γ e 3 3 n e 2 k 2 R ) × exp ( γ e R n e 2 k 2 β 2 ) | G ν β | 2 d β .
2 α = π A e 2 2 s P a exp ( 4 Δ w 3 3 a υ 2 R ) w ( w R a + υ 2 2 Δ w ) 1 / 2 .
υ = k a n m ( 2 Δ ) 1 / 2 .
β 0 a = 1 / ( G 2 w υ 2 + 1 w ) 1 / 2 ,
n ( r ) = n 0 { 1 Δ [ 1 1 2 erf ( a + r 2 D ) 1 2 erf ( a r 2 D ) ] } ,
erf ( x ) = 2 π 0 x exp ( z 2 ) d z .
Δ R * / a A ( υ ) or Δ 3 / 2 R * / λ B ( υ )
A e 2 / P = C ( υ ) / a 2 .
R 2 * R 1 * = h m 3 / 2 [ 1 + C R ( 1 h m ) + 1 2 ( h m ) 2 C R 2 ( 1 m h ) ] ,
h = λ 2 / λ 1 m = Δ 2 / Δ 1 ,
C R = 3 υ 3 λ 1 8 2 π n m w 3 Δ 1 3 / 2 R 1 * ,
a 2 / a 1 = h / m 1 / 2 .
R * ( λ 1 , Δ 1 ; υ ) R * ( λ 1 , Δ 1 ; υ ) = R * ( λ 2 , Δ 2 ; υ ) R * ( λ 2 , Δ 2 ; υ ) .
q 11 = ( π κ i r 1 / 2 ) [ J ν + 1 ( 2 ) N ν ( 1 ) J ν ( 1 ) N ν + 1 ( 2 ) ] , q 12 = ( π κ i 2 r 1 / 2 β g ) [ J ν + 1 ( 2 ) N ν + 1 ( 1 ) J ν + 1 ( 1 ) N ν + 1 ( 2 ) ] , q 21 = ( π β g r 1 / 2 ) [ J ν ( 2 ) N ν ( 1 ) J ν ( 1 ) N ν ( 2 ) ] , q 22 = ( π κ i r 1 / 2 ) [ J ν ( 2 ) N ν + 1 ( 1 ) J ν + 1 ( 1 ) N ν ( 2 ) ] . }
q 11 = γ i r 1 [ I ν + 1 ( 2 ) K ν ( 1 ) + I ν ( 1 ) K ν + 1 ( 2 ) ] , q 12 = ( γ i 2 r 1 / β g ) [ I ν + 1 ( 2 ) K ν + 1 ( 1 ) I ν + 1 ( 1 ) K ν + 1 ( 2 ) ] , q 21 = β g r 1 [ I ν ( 2 ) K ν ( 1 ) I ν ( 1 ) K ν ( 2 ) ] , q 22 = γ i r 1 [ I ν ( 2 ) K ν + 1 ( 1 ) + I ν + 1 ( 1 ) K ν ( 2 ) ] . }
[ z y ] r = a 1 = U [ κ 0 J ν + 1 ( κ 0 a 1 ) / β g J ν ( κ 0 a 1 ) ] A 1 ,
[ z y ] r = a 1 = U [ γ 0 I ν + 1 ( γ 0 a 1 ) / β g I ν ( γ 0 a 1 ) ] B 1 .
[ z y ] r = a N = U [ γ e K ν + 1 ( γ e a N ) / β g K ν ( γ e a N ) ] A e .
[ γ e K ν + 1 ( γ e a N ) / β g , f 11 κ 0 J ν + 1 ( κ 0 a 1 ) / β g f 12 J ν ( κ 0 a 1 ) K ν ( γ e a N ) , f 21 κ 0 J ν + 1 ( κ 0 a 1 ) / β g f 22 J ν ( κ 0 a 1 ) ] × [ A 1 A e ] = 0.
Z ν 2 ( α h ) r d r = ( r 2 / 2 ) [ Z ν 2 ( α h ) Z ν 1 ( α h ) Z ν + 1 ( α h ) ] ,
J ν ( α h ) N ν ( α h ) r d r = ( r 2 / 2 ) [ J ν ( α h ) N ν ( α h ) J ν 1 ( α h ) N ν + 1 ( α h ) ] . K ν ( α h ) I ν ( α h ) r d r = ( r 2 / 2 ) [ I ν ( α h ) K ν ( α h ) + K ν 1 ( α h ) I ν + 1 ( α h ) ] .
G 0 β = π exp [ a ( γ e 2 + β 2 ) 1 / 2 ] ( γ e 2 + β 2 ) 1 / 2 , G 1 β = ( π / γ e ) exp [ a ( γ e 2 + β 2 ) 1 / 2 ] .
P clad / P = ( A e 2 a 2 / P ) [ K ν 1 ( w ) K ν + 1 ( w ) K ν ) w ) 2 ] .
R * ( λ 1 , Δ 1 ; 2 α 2 ) R * ( λ 1 , Δ 1 ; 2 α 1 ) = 1 C R ln ( α 2 / α 1 ) .

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